Suppose that we want to count the partitions of $n$ into distinct parts no larger than $4$. For each of the sizes $1,2,3$, and $4$ we can have either $0$ or $1$ part of that size, so the generating function is $(1+x)(1+x^2)(1+x^3)(1+x^4)$: e.g., the term $1\cdot x^2\cdot1\cdot x^4$, for instance, corresponds to the partition $2+4$ of $6$. Thus, if $p_d(n)$ is the number of partitions of $n$ into distinct parts, we must have

$$\sum_{n\ge 0}p_d(n)x^n=\prod_{k\ge 1}\left(1+x^k\right)\;.$$

How should you modify this to get the generating function for the number of partitions of $n$ into distinct odd parts?